## 講演会

### 2007年01月10日(水)

16:00-17:30   数理科学研究科棟(駒場) 118号室
Oleg Yu. Emanouilov 氏 (Colorado State University)
Some Problems of Global Controllability of Burgers Equation and Navier-Stokes system.

### 2007年01月09日(火)

16:00-17:30   数理科学研究科棟(駒場) 118号室

「魅力ある大学院教育」イニシアティブにより以下の講演を行います。

Oleg Yu. Emanouilov 氏 (Colorado State University)
Some Problems of Global Controllability of Burgers Equation and Navier-Stokes system.
[ 講演概要 ]
We show that 1-D Burgers equation is globally uncontrollable with control acting at two endpoints. Then we establish the global controllability of the 2-D Burgers equation. Finally we show that for 2-D Navier-Stokes system the problem of global exact controllability is solvable for the dense set of the initial data with a control acting on part of the boundary.

### 2006年12月08日(金)

10:30-12:00   数理科学研究科棟(駒場) 056号室
Charles M. Elliott 氏 (University of Sussex)
Computational Methods for Surface Partial Differential Equations
[ 講演概要 ]
In these lectures we discuss the formulation, approximation and applications of partial differential equations on stationary and evolving surfaces. Partial differential equations on surfaces occur in many applications. For example, traditionally they arise naturally in fluid dynamics, materials science, pattern formation on biological organisms and more recently in the mathematics of images. We will derive the conservation law on evolving surfaces and formulate a number of equations.

We propose a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\\Gamma$ in $\\mathbb R^{n+1}$. The key idea is based on the approximation of $\\Gamma$ by a polyhedral surface $\\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. We extend this approach to pdes on evolving surfaces. We define an Eulerian level set method for partial differential equations on surfaces. The key idea is based on formulating the partial differential equation on all level set surfaces of a prescribed function $\\Phi$ whose zero level set is $\\Gamma$. We use Eulerian surface gradients to define weak forms
of elliptic operators which naturally generate weak formulations
of Eulerian elliptic and parabolic equations. This results in a degenerate equation formulated in anisotropic Sobolev spaces based on the level set function $\\Phi$. The resulting equation is then solved in one space dimension higher but can be solved on a fixed finite element grid.

Numerical experiments are described for several linear and Nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow. In particular we show how surface level set and phase field models can be used to compute the motion of curves on surfaces. This is joint work with G. Dziuk(Freiburg).

### 2006年12月07日(木)

13:00-14:30   数理科学研究科棟(駒場) 056号室

Charles M. Elliott 氏 (University of Sussex)
Computational Methods for Surface Partial Differential Equations
[ 講演概要 ]
In these lectures we discuss the formulation, approximation and applications of partial differential equations on stationary and evolving surfaces. Partial differential equations on surfaces occur in many applications. For example, traditionally they arise naturally in fluid dynamics, materials science, pattern formation on biological organisms and more recently in the mathematics of images. We will derive the conservation law on evolving surfaces and formulate a number of equations.

We propose a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\\Gamma$ in $\\mathbb R^{n+1}$. The key idea is based on the approximation of $\\Gamma$ by a polyhedral surface $\\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. We extend this approach to pdes on evolving surfaces. We define an Eulerian level set method for partial differential equations on surfaces. The key idea is based on formulating the partial differential equation on all level set surfaces of a prescribed function $\\Phi$ whose zero level set is $\\Gamma$. We use Eulerian surface gradients to define weak forms
of elliptic operators which naturally generate weak formulations
of Eulerian elliptic and parabolic equations. This results in a degenerate equation formulated in anisotropic Sobolev spaces based on the level set function $\\Phi$. The resulting equation is then solved in one space dimension higher but can be solved on a fixed finite element grid.

Numerical experiments are described for several linear and Nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow. In particular we show how surface level set and phase field models can be used to compute the motion of curves on surfaces. This is joint work with G. Dziuk(Freiburg).
[ 参考URL ]
https://www.u-tokyo.ac.jp/campusmap/map02_02_j.html

### 2006年12月01日(金)

16:00-18:00   数理科学研究科棟(駒場) 126号室

von Neumann 環上の群作用
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

### 2006年11月30日(木)

16:00-18:00   数理科学研究科棟(駒場) 126号室

von Neumann 環上の群作用
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

### 2006年11月29日(水)

16:00-18:00   数理科学研究科棟(駒場) 122号室

von Neumann 環上の群作用
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

### 2006年11月28日(火)

16:00-18:00   数理科学研究科棟(駒場) 122号室

von Neumann 環上の群作用
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

### 2006年11月27日(月)

16:00-18:00   数理科学研究科棟(駒場) 122号室

von Neumann 環上の群作用
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

### 2006年11月16日(木)

16:30-18:00   数理科学研究科棟(駒場) 118号室
Pierre Berthelot 氏 (Rennes大学)
Crystalline complexes and D-modules

### 2006年11月15日(水)

16:30-18:00   数理科学研究科棟(駒場) 117号室
Pierre Berthelot 氏 (Rennes大学)
Crystalline complexes and D-modules

### 2006年11月09日(木)

16:20-17:50   数理科学研究科棟(駒場) 123号室
S. Bloch 氏 (シカゴ大学)
<連続講演> Graphs and motives

### 2006年11月08日(水)

16:20-17:50   数理科学研究科棟(駒場) 123号室
S. Bloch 氏 (シカゴ大学)
<連続講演> Graphs and motives

### 2006年11月07日(火)

16:20-17:50   数理科学研究科棟(駒場) 123号室
11月7日(火)-11月9日(木)の連続講演です.

S. Bloch 氏 (シカゴ大学)
<連続講演> Graphs and motives

### 2006年10月25日(水)

16:30-18:00   数理科学研究科棟(駒場) 122号室
Heinz W. Engl 氏 (Industrial Mathematics Institute, Kepler University, Linz and Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences)
Regularization of nonlinear inverse problems: mathematics, industrial application fields, new challenges
[ 講演概要 ]
Motivated by some of the industrial examples presented in the first talk, we outline the theory of regularization methods for the stable solution of nonlinear inverse problems. Then, we turn to some new problem fields of possible future industrial relevance in systems and molecular biology.

### 2006年10月23日(月)

16:30-18:00   数理科学研究科棟(駒場) 122号室
Heinz W. Engl 氏 (Industrial Mathematics Institute, Kepler University, Linz and Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences )
Mathematical modelling and numerical simulation: from iron and steel making via inverse problems to finance
[ 講演概要 ]
We first describe the industrial mathematics structure in linz, extending from basic research via graduate education to industrial collaboration. We then present a few projetcs from our experience, ranging from aspects of iron and steel processing via mathematical simulation and optimization in car industry to robust and fast pricing methods for financial derivates. Since some of the projects involve inverse problem, we give a first introduction into this field, which will be deepened in the second talk.